The square root of `(7 + 3sqrt5) (7 - 3sqrt5) ` is :

Options

A. 4
B. `sqrt5`
C. `3sqrt5`
D. 2

Answer

To solve the problem of finding the square root of \( (7 + 3\sqrt{5})(7 - 3\sqrt{5}) \), we can follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ (7 + 3\sqrt{5})(7 - 3\sqrt{5}) \]

### Step 2: Recognize the formula
This expression is in the form of \( (a + b)(a - b) \), which can be simplified using the difference of squares formula:
\[ a^2 - b^2 \]
Here, \( a = 7 \) and \( b = 3\sqrt{5} \).

### Step 3: Calculate \( a^2 \) and \( b^2 \)
Now we will calculate \( a^2 \) and \( b^2 \):
\[ a^2 = 7^2 = 49 \]
\[ b^2 = (3\sqrt{5})^2 = 3^2 \cdot (\sqrt{5})^2 = 9 \cdot 5 = 45 \]

### Step 4: Substitute back into the difference of squares formula
Now substitute \( a^2 \) and \( b^2 \) into the difference of squares formula:
\[ (7 + 3\sqrt{5})(7 - 3\sqrt{5}) = a^2 - b^2 = 49 - 45 \]

### Step 5: Simplify the expression
Now simplify the expression:
\[ 49 - 45 = 4 \]

### Step 6: Take the square root
Finally, we take the square root of the result:
\[ \sqrt{4} = 2 \]

### Conclusion
Thus, the square root of \( (7 + 3\sqrt{5})(7 - 3\sqrt{5}) \) is:
\[ \boxed{2} \]